The general iterative methods for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings in Hilbert spaces

Kiattisak Rattanaseeha

doi:10.1186/1029-242x-2013-153

Abstract

In this paper, the researcher introduces the general iterative scheme for finding a common element of the set of equilibrium problems and fixed point problems of a countable family of nonexpansive mappings in Hilbert spaces. The results presented in this paper improve and extend the corresponding results announced by many others. MSC: 47H10; 47H09

Highlights

Let H be a real Hilbert space with the inner product ·, · and the norm ·
Let F be a bifunction from C × C into R, where R is the set of real numbers
Optimization and economics reduce to finding a solution of ( . ). ( ) The mappings {Tn}n∈N are said to be a family of nonexpansive mappings from H into itself if

Summary

Introduction

Let H be a real Hilbert space with the inner product ·, · and the norm ·. Marino and Xu [ ] introduced the following general iterative method: xn+ = (I – αnA)Txn + αnγ f (xn), n ≥ , where A is a strongly positive bounded linear operator on H. They proved that if the sequence {αn} of parameters satisfies appropriate conditions, the sequence {xn} generated by ) converges strongly to the unique solution of the variational inequality (A – γ f )x∗, x – x∗ ≥ , x ∈ C, which is the optimality condition for the minimization problem min Ax, x – h(x), x∈C where h is a potential function for γ f (i.e., h (x) = γ f (x) for x ∈ H). {Tn}∞ n= be a sequence of nonexpansive mappings from C to C such that the common fixed point set (Tn)

Let f

Note that

We derive then